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Sato-Tate Equidistribution for Families of Hecke-Maass Forms on SL (N, R)/SO

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Sato-Tate Equidistribution for Families of Hecke-Maass Forms on SL (N, R)/SO SATO-TATE EQUIDISTRIBUTION FOR FAMILIES OF HECKE–MAASS FORMS ON SL(n, R)/ SO(n) JASMIN MATZ AND NICOLAS TEMPLIER Abstract. We establish the Sato-Tate equidistribution of Hecke eigenvalues of the family of Hecke–Maass cusp forms on SL(n, Z) SL(n, R)/ SO(n). As part of the proof, we establish a uniform upper-bound for spherical functions on\ semisimple Lie groups which is of independent interest. For each of the principal, symmetric square and exterior square L-functions, we deduce the level dis- tribution with restricted support of the low-lying zeros. We also deduce average estimates toward Ramanujan, including an improvement on the previous literature in the case n = 2. Contents 1. Introduction 1 2. Symmetry type of families and low-lying zeros 12 3. Average bounds towards Ramanujan 15 Part 1. Local theory: Real orbital integrals 18 4. Preliminaries 19 5. Setting for the main estimate 23 6. Bounds for semisimple orbital integrals 26 7. Weighted orbital integrals 29 8. Spherical functions and archimedean test functions 39 Part 2. Global theory: Weyl’s law and equidistribution 44 9. Setup for the Arthur-Selberg trace formula for GL(n) 44 10. Centralizers of semisimple elements 45 11. Coarse and fine geometric expansion 50 12. Bound for p-adic weighted orbital integrals 61 13. Spectral side and conclusion of proof of Theorem 1.1 67 References 72 arXiv:1505.07285v8 [math.NT] 18 May 2020 1. Introduction Hecke–Maass cusp forms are certain eigenfunctions of the Laplace operator on the locally sym- metric space SL(n, Z) SL(n, R)/ SO(n). Beyond the existence of such forms and structure theory, we want to study spectral\ properties such as the Weyl’s law, the distribution of Hecke eigenvalues, temperedness, and average behavior in families. Major difficulties in the analysis of the trace for- mula arise when the space is not compact, and when the test function is not of compact support. In this paper we deal with both difficulties together and solve a long-standing equidistribution problem that generalizes some classical results of Selberg for n = 2. Date: May 20, 2020. Key words and phrases. Automorphic forms, L-functions. 1 2 Selberg [Sel56] introduced the trace formula and derived the Weyl’s law for SL(2, Z), which is an asymptotic count for the family of Hecke–Maass cusp forms on SL(2, Z) SL(2, R)/ SO(2), ordered by the size of their eigenvalues. Much later, Sarnak [Sar87] observed that\ a variant of the same argument can be used to establish a much more precise result concerning averages of Hecke eigenvalues, namely the Sato-Tate equidistribution for the same family of Hecke–Maass cusp forms. This entails to inserting test functions of varying support in the Selberg trace formula, such as a Hecke operator Tp with p growing arbitrary large, and estimating the geometric side in a concrete way. Sarnak and Piatetskii-Shapiro [Sar87, 4] then raised the problem of generalizing this to the family of Hecke–Maass cusp forms on SL(§n, Z) SL(n, R)/ SO(n) for an arbitrary n > 2. The present paper solves this problem. \ Arthur generalized [Art78] the Selberg trace formula to general reductive groups, by decomposing the geometric side into coarse equivalence classes and constructing truncation operators. He further introduced in [Art81] the weighted orbital integrals, which enabled him to develop in [Art86b] a fine expansion of the geometric side. Especially important for our purposes are the splitting formulas established in [Art81, 6], and the weighted measures constructed in [Art88a]. In addition to these§ fundamental results of Arthur, we use the recursive analysis by Lapid– M¨uller [LM09] of the spectral side of the trace formula for GL(n), the method and results by Shin and the second-named author [ST16] on uniform estimates of certain orbital integrals, and the estimate by the first-named author [Mat15] of Arthur’s global coefficients. Then there are two important novelties. We develop in Part 1 some uniform germ estimates for orbital integrals of certain bi-SO(n)-invariant unbounded functions. This difficulty arises because the test function is not of compact support, and our method is of independent interest, notably in view of the new estimates on Harish-Chandra spherical functions. In Part 2, we establish uniform bounds for all the terms that appear in Arthur’s fine geometric expansion for GL(n). This paper is also the first to establish a remainder term in the Weyl’s law for SL(n, Z) when n > 3. 1.1. Main result. Let G = GL(n) and A = R Af the ring of adeles of Q. Let K = K Kf the usual maximal compact subgroup of × ∞ · G(A)1 := g G(A), det g =1 { ∈ | |A } given by K = O(n) G(R) and Kf = G(Zˆ) G(Af ). Let K◦ = SO(n) K be the identity ∞ ⊆ 1 ⊆ ∞ ⊆ ∞ component of K . Let Πcusp(G(A) ) denote the set of irreducible unitary representations π oc- curring in the cuspidal∞ part of L2(G(Q) G(A)1). Such π can be uniquely extended to a cuspidal automorphic representation of G(A) whose\ central character has finite order, and conversely. We say that π is spherical (resp. unramified) if πK∞ (resp. πKf ) is non-zero. Unramified represen- 1 tations π in Πcusp(G(A) ) correspond to unramified cuspidal automorphic representation of G(A) with trivial central character. 1 For π Πcusp(G(A) ) let λπ aC∗ /W denote the infinitesimal character of the archimedean component∈ π . Here a is the Lie∈ algebra of the subgroup A G(R)1 of diagonal matrices with ∞ ⊂ positive entries and W S is the Weyl group. For t> 0 and a bounded open subset Ω ia∗ let ≃ n ⊆ 2 1 1 c(ρ) Λ (t) := 2 vol(G(Q) G(A) /K ) W − dλ Ω \ f | | c(λ) ZtΩ 1 where c denotes Harish-Chandra’s c-function for G(R) := g G (R) det g R =1 , and ρ is the d { ∈ || 1 | } n(n+1) half-sum of positive roots. It is of order t as t where d = dimR(G(R) /K )= 1. →∞ ∞ 2 − We define characters χ : O(n)/ Id = PO(n) 1 as follows: χ+ is the trivial character, ± {± } −→ {± } that is, χ+(k) = 1 for all k PO(n), and χ (k) = det k if n is even and is the trivial character if ∈ − 3 n is odd. Note that the group PO(n) is disconnected if n is even in which case det( Id) = 1 while PO(n) is connected if n is odd in which case det( Id) = 1 . We view{ χ± as} unitary{ } characters on K = O(n) which are invariant under{ O(n±) Z}(R){± = } Id , and which± are both trivial if n is odd.∞ Here Z denotes the center of G. ∩ {± } If χ χ+, χ then π χ defines another element in the unitary dual of G(R), and the set of fixed∈ vectors { − (π} χ)K∞∞⊗under K is non-empty if and only if π has K -type χ, that is, if ∞ ⊗ ∞ ∞ ∞ K∞ K∞ one vector in π is χ-invariant with respect to the K -action. If χ = χ+, (π χ+) = π =0 means that π ∞is spherical. ∞ ⊗ 6 ∞ Theorem 1.1. For any integer n > 3 there is an effective constant A > 0 depending only on 2 n, and for any non-empty W -invariant bounded open subset Ω ia∗ with piecewise C -boundary ⊆ there is a constant c1 > 0 such that the following holds. Let χ χ+, χ and let τ : G(Af ) C ∈{ −} −→ be the characteristic function of a compact bi-Kf -invariant subset. Then 1 K∞ lim ΛΩ(t)− dim(π χ) tr πf (τ)= τ(γ), t ∞ 1 ⊗ →∞ π Πcusp(G(A) ), γ Z(Q)/ 1 ∈ ∈ {± } λXπ tΩ X ∈ where Z is the center of G. Moreover, for all t > 1 we have K∞ A d 1/2 dim(π χ) tr πf (τ) ΛΩ(t) τ(γ) 6 c1 τ L1(G(A ))t − . ∞ f 1 ⊗ − k k π Πcusp(G(A) ), γ Z(Q)/ 1 ∈ ∈ {± } λXπ tΩ X ∈ Example 1.2. If τ = τ0 is the characteristic function of Kf , then Theorem 1.1 is a Weyl’s law with remainder term, 1 K∞ Kf d 1/2 (1.1) π Πcusp(G(A) ), λπ tΩ, (π χ) =0, πf =0 =ΛΩ(t)+ O(t − ), { ∈ ∈ ∞ ⊗ 6 6 } which was also established in an unpublished manuscript of the second -named author. This is new already for n = 3. The asymptotic Weyl’s law, i.e., without remainder term, was established for SL3(Z) by Miller [Mil01], for SLn(Z), n > 3, by M¨uller [M¨ul07], and for quasi-split reductive groups by Lindenstrauss–Venkatesh [LV07]. Remark 1.3. Conditional on the assumption that the lattice G(Q) Kf is neat, which is false in our case since PSL(n, Z) is not torsion free, a stronger version of the∩ Weyl’s law (1.1) with remainder d 1 max(3,n) t − (log t) is due to Lapid–M¨uller [LM09]. 1.2. Hecke–Maass forms. We can restate the result classically in terms of Hecke–Maass cusp forms which are smooth functions f on 1 SLn(Z) SLn(R)/ SOn(R) G(Q) G(A) /K◦ Kf \ ≃ \ ∞ that are eigenfunctions of the Laplace operator, the Hecke operators and are cuspidal. Hecke– Maass cusp forms can be divided into even and odd forms. Let W denote the Hecke operator corresponding to the double coset SLn(Z) diag( 1, 1,..., 1)SLn(Z) = SLn(Z) diag( 1, 1,..., 1).
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